In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its centre. It is a manifold of codimension one—that is, with one dimension less than that of the ambient space.

As the hypersphere's radius increases, its curvature decreases. In the limit, a hypersphere approaches the zero curvature of a hyperplane. Hyperplanes and hyperspheres are examples of hypersurfaces.

The term hypersphere was introduced by Duncan Sommerville in his discussion of models for non-Euclidean geometry.[1] The first one mentioned is a 3-sphere in four dimensions.

Some spheres are not hyperspheres: If S is a sphere in Em where m < n, and the space has n dimensions, then S is not a hypersphere. Similarly, any n-sphere in a proper flat is not a hypersphere. For example, a circle is not a hypersphere in three-dimensional space, but it is a hypersphere in the plane.


  1. Sommerville, D. M. Y. (1914). "'Space Curvature' and the Philosophical Bearing of Non-Euclidean Geometry" (PDF). In Milne, William P. (ed.). The Elements of Non-Euclidean Geometry. Bell's Mathematical Series for Schools and Colleges. London: G. Bell and Sons. p. 193 via University of Michigan Historical Math Collection.

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